I would love to auto start Bartender (by the way, excellent app that hides some of your notification area icons) when I unplug my external monitor from my Mac.
Is there any app or script that could do that?
Try ControlPlane. It allows you to trigger various actions, including launching applications, based on a variety of "evidence sources" (which include which monitors are connected). Have a look at the full feature list, but it should accomplish what you're looking for.